Parametrization of a singular Lagrangian variety

Ishikawa, Goo

doi:10.1090/S0002-9947-1992-1044961-9

Parametrization of a singular Lagrangian variety
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Trans. Amer. Math. Soc. 331 (1992), 787-798 Request permission

Abstract:

We give stabilization and parametrization theorems for a class of singular varieties in the space of polynomials of one variable and generalize the results of Arnol’d and Givental’. The class contains the open swallowtails and the open Whitney umbrella. The parametrization is associated with the singularity of a stable mapping (in the sense of Thom and Mather) of kernel rank one.

References

Foundation of mechanics

Stephanie B. Alexander, I. David Berg, and Richard L. Bishop, Cauchy uniqueness in the Riemannian obstacle problem, Differential geometry, Peñíscola 1985, Lecture Notes in Math., vol. 1209, Springer, Berlin, 1986, pp. 1–7. MR 863742, DOI 10.1007/BFb0076617
V. I. Arnol′d, Lagrangian manifolds with singularities, asymptotic rays and the unfurled swallowtail, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 1–14, 96 (Russian). MR 639196

Singularities in variational calculus

22

27

Th. Bröcker, Differentiable germs and catastrophes, London Mathematical Society Lecture Note Series, No. 17, Cambridge University Press, Cambridge-New York-Melbourne, 1975. Translated from the German, last chapter and bibliography by L. Lander. MR 0494220
A. B. Givental′, Manifolds of polynomials that have a root of fixed comultiplicity, and the generalized Newton equation, Funktsional. Anal. i Prilozhen. 16 (1982), no. 1, 13–18, 96 (Russian). MR 648804

Lagrangian imbeddings of surfaces and unfolded Whitney umbrella

20

M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518
Goo Ishikawa, Families of functions dominated by distributions of ${\cal C}$-classes of mappings, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 199–217. MR 699495
Stanisław Janeczko, Generating families for images of Lagrangian submanifolds and open swallowtails, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 91–107. MR 838655, DOI 10.1017/S0305004100065889
Stanisław Janeczko, Constrained Lagrangian submanifolds over singular constraining varieties and discriminant varieties, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 1–26 (English, with French summary). MR 877993
Stanisław Janeczko, Tensor invariants and invariant symplectic geometry of binary forms, Bull. Polish Acad. Sci. Math. 36 (1988), no. 1-2, 15–23 (English, with Russian summary). MR 998203
Joseph B. Keller, A geometrical theory of diffraction, Calculus of variations and its applications. Proceedings of Symposia in Applied Mathematics, Vol. VIII, McGraw-Hill Book Co., Inc., New York-Toronto-London, for the American Mathematical Society, Providence, R.I., 1958, pp. 27–52. MR 0094120
B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
B. Malgrange, Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math. 46 (1976), 163–173 (French). MR 508169

Notes on topological stability

A. N. Varchenko and A. B. Givental′, The period mapping and the intersection form, Funktsional. Anal. i Prilozhen. 16 (1982), no. 2, 7–20, 96 (Russian). MR 659161
Alan Weinstein, Lectures on symplectic manifolds, Regional Conference Series in Mathematics, No. 29, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976. MR 0464312
Hassler Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496–549. MR 192520, DOI 10.2307/1970400